In geometry and topology, there are certain structures that are very "rigid" (like Riemannian manifolds) and others that are very "flexible" (like topological manifolds). Symplectic and contact geometry lies in between these two extremes and incorporates some attractive features of both. One consequence is that symplectic techniques have recently been used, to great effect, to give combinatorial approaches to questions in topology that were previously only amenable to difficult gauge-theoretic and analytic techniques. I will introduce symplectic and contact structures and describe some recent developments linking them to the study of three-dimensional manifolds and knots.